Subgroups — C₃₁×#3339 #16843272

Order 3968.

366 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#553	C₂	1	2	yes	maximal, complemented
2	C₃₁×#499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#482	C₂	1	2	yes	maximal, complemented
4	C₃₁×#481	C₂	1	2	yes	maximal, complemented
5	C₃₁×#478	C₂	1	2	yes	maximal, complemented
6	C₃₁×#466	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#466	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#427	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#107	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#126	—	2	2	no	—
16	C₃₁×#125	C₂²	1	2	yes	—
17	C₃₁×#123	—	2	2	no	—
18	C₃₁×#121	—	2	2	no	—
19	C₃₁×#121	C₂²	1	2	yes	—
20	C₃₁×#121	C₂²	1	2	yes	—
21	C₄×D₄×C₃₁	—	2	2	no	—
22	C₄×D₄×C₃₁	C₂²	1	2	yes	—
23	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#117	C₂²	1	2	yes	—
25	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₃₁×#107	C₂²	1	1	yes	characteristic
27	C₃₁×#107	C₂²	1	1	yes	characteristic
28	C₃₁×#107	C₂²	1	2	yes	—
29	C₃₁×#107	C₂²	1	1	yes	characteristic
30	C₃₁×#106	C₂²	1	2	yes	—
31	C₃₁×#106	C₂²	1	2	yes	—
32	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₃₁×#102	C₂²	1	2	yes	—
34	C₃₁×#102	C₂²	1	2	yes	—
35	C₃₁×#102	C₂²	1	2	yes	—
36	C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
37	C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
39	C₂×D₄×C₃₁	—	2	2	no	—
40	C₂×D₄×C₃₁	—	2	2	no	—
41	C₂×D₄×C₃₁	D₄	1	1	yes	characteristic
42	C₂×D₄×C₃₁	C₂³	1	2	yes	—
43	C₂²×C₄×C₃₁	—	2	2	no	—
44	C₂²×C₄×C₃₁	—	2	2	no	—
45	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
47	D₈×C₃₁	—	2	2	no	—
48	C₂×C₈×C₃₁	—	2	2	no	—
49	C₂×C₈×C₃₁	—	2	2	no	—
50	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₃₁	—	2	2	no	—
55	C₄⋊C₄×C₃₁	—	2	2	no	—
56	C₄⋊C₄×C₃₁	—	2	2	no	—
57	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
58	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
59	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
60	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
61	C₂²⋊C₄×C₃₁	—	2	2	no	—
62	C₂²⋊C₄×C₃₁	—	2	2	no	—
63	C₂²⋊C₄×C₃₁	—	2	2	no	—
64	C₂²⋊C₄×C₃₁	—	2	2	no	—
65	C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
66	C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
67	C₄²×C₃₁	—	2	2	no	—
68	C₄²×C₃₁	C₂³	1	1	yes	characteristic
69	C₂³×C₃₁	—	2	2	no	—
70	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
72	D₄×C₃₁	—	2	2	no	—
73	D₄×C₃₁	—	2	2	no	—
74	D₄×C₃₁	—	2	2	no	—
75	D₄×C₃₁	—	4	4	no	—
76	D₄×C₃₁	—	8	2	no	—
77	C₂×C₄×C₃₁	—	2	1	no	—
78	C₂×C₄×C₃₁	—	2	1	no	—
79	C₂×C₄×C₃₁	—	2	1	no	—
80	C₂×C₄×C₃₁	—	2	1	no	—
81	C₂×C₄×C₃₁	—	2	2	no	—
82	C₂×C₄×C₃₁	—	2	2	no	—
83	C₂×C₄×C₃₁	—	2	2	no	—
84	C₂×C₄×C₃₁	—	8	2	no	—
85	C₂×C₄×C₃₁	—	8	2	no	—
86	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
91	C₈×C₃₁	—	2	1	no	—
92	C₈×C₃₁	—	2	1	no	—
93	C₈×C₃₁	—	2	1	no	—
94	C₈×C₃₁	#41	1	2	yes	—
95	#3339	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂²×C₃₁	—	2	1	no	—
97	C₂²×C₃₁	—	2	1	no	—
98	C₂²×C₃₁	—	2	1	no	—
99	C₂²×C₃₁	—	2	1	no	—
100	C₂²×C₃₁	—	2	1	no	—
101	C₂²×C₃₁	—	4	2	no	—
102	C₂²×C₃₁	—	4	2	no	—
103	C₂²×C₃₁	—	4	2	no	—
104	C₂²×C₃₁	#135	1	2	yes	—
105	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
106	C₄×C₃₁	—	2	1	no	—
107	C₄×C₃₁	—	2	1	no	—
108	C₄×C₃₁	—	2	1	no	—
109	C₄×C₃₁	—	2	1	no	—
110	C₄×C₃₁	—	4	2	no	—
111	C₄×C₃₁	—	4	2	no	—
112	C₄×C₃₁	—	4	2	no	—
113	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
114	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
115	#553	C₂×C₃₁	1	2	yes	complemented
116	#499	C₂×C₃₁	1	1	yes	characteristic, complemented
117	#482	C₂×C₃₁	1	2	yes	complemented
118	#481	C₂×C₃₁	1	2	yes	complemented
119	#478	C₂×C₃₁	1	2	yes	complemented
120	#466	C₂×C₃₁	1	1	yes	characteristic, complemented
121	#466	C₂×C₃₁	1	1	yes	characteristic, complemented
122	C₄×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
123	C₈×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
124	#427	C₂×C₃₁	1	1	yes	characteristic, complemented
125	C₂×#107	C₂×C₃₁	1	1	yes	characteristic, complemented
126	C₂×C₃₁	—	2	2	no	—
127	C₂×C₃₁	—	4	1	no	—
128	C₂×C₃₁	—	8	2	no	—
129	C₂×C₃₁	#547	1	1	yes	characteristic
130	C₂×C₃₁	C₂×#135	1	1	yes	characteristic
131	C₂×C₃₁	#576	1	1	yes	characteristic
132	C₂×D₈	C₂²×C₃₁	1	1	yes	characteristic
133	C₂²×C₈	C₂²×C₃₁	1	1	yes	characteristic
134	C₂²×C₈	C₂²×C₃₁	1	1	yes	characteristic
135	#126	—	2	2	no	—
136	#125	C₂²×C₃₁	1	2	yes	—
137	#123	—	2	2	no	—
138	#121	—	2	2	no	—
139	#121	C₂²×C₃₁	1	2	yes	—
140	#121	C₂²×C₃₁	1	2	yes	—
141	C₄×D₄	—	2	2	no	—
142	C₄×D₄	C₂²×C₃₁	1	2	yes	—
143	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
144	#117	C₂²×C₃₁	1	2	yes	—
145	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
146	#107	C₂²×C₃₁	1	1	yes	characteristic
147	#107	C₂²×C₃₁	1	1	yes	characteristic
148	#107	C₂²×C₃₁	1	2	yes	—
149	#107	C₂²×C₃₁	1	1	yes	characteristic
150	#106	C₂²×C₃₁	1	2	yes	—
151	#106	C₂²×C₃₁	1	2	yes	—
152	C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
153	#102	C₂²×C₃₁	1	2	yes	—
154	#102	C₂²×C₃₁	1	2	yes	—
155	#102	C₂²×C₃₁	1	2	yes	—
156	C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
157	C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
158	C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
159	C₃₁	#3339	1	1	yes	characteristic, direct product factor, Sylow
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	—	2	2	no	—
162	C₂×D₄	D₄×C₃₁	1	1	yes	characteristic
163	C₂×D₄	C₂³×C₃₁	1	2	yes	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
167	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
168	D₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
172	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
173	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
174	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
179	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
180	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
181	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	2	2	no	—
186	C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
187	C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
188	C₄²	—	2	2	no	—
189	C₄²	C₂³×C₃₁	1	1	yes	characteristic
190	C₂³	—	2	2	no	—
191	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
192	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
193	D₄	—	2	2	no	—
194	D₄	—	2	2	no	—
195	D₄	—	2	2	no	—
196	D₄	—	4	4	no	—
197	D₄	—	8	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	8	2	no	—
206	C₂×C₄	—	8	2	no	—
207	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
208	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
209	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
210	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
211	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
212	C₈	—	2	1	no	—
213	C₈	—	2	1	no	—
214	C₈	—	2	1	no	—
215	C₈	#41×C₃₁	1	2	yes	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	—	4	2	no	—
224	C₂²	C₃₁×#135	1	2	yes	—
225	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	4	2	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	2	no	—
233	C₄	C₂×#41×C₃₁	1	1	yes	characteristic
234	C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
235	C₂	—	2	2	no	—
236	C₂	—	4	1	no	—
237	C₂	—	8	2	no	—
238	C₂	C₃₁×#547	1	1	yes	characteristic
239	C₂	C₂×C₃₁×#135	1	1	yes	characteristic
240	C₂	C₃₁×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×D₄×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃₁×#553	C₂	semidirect
2. C₃₁×#499	C₂	semidirect
3. C₃₁×#482	C₂	semidirect
4. C₃₁×#481	C₂	semidirect
5. C₃₁×#478	C₂	semidirect
6. C₃₁×#466	C₂	semidirect
7. C₃₁×#466	C₂	semidirect
8. C₄×D₈×C₃₁	C₂	semidirect
9. C₈×D₄×C₃₁	C₂	semidirect
10. C₃₁×#427	C₂	semidirect
11. C₂×C₃₁×#107	C₂	semidirect
95. #3339	C₃₁	direct (N × H)
115. #553	C₂×C₃₁	semidirect
116. #499	C₂×C₃₁	semidirect
117. #482	C₂×C₃₁	semidirect
118. #481	C₂×C₃₁	semidirect
119. #478	C₂×C₃₁	semidirect
120. #466	C₂×C₃₁	semidirect
121. #466	C₂×C₃₁	semidirect
122. C₄×D₈	C₂×C₃₁	semidirect
123. C₈×D₄	C₂×C₃₁	semidirect
124. #427	C₂×C₃₁	semidirect
125. C₂×#107	C₂×C₃₁	semidirect
159. C₃₁	#3339	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#553	2
C₃₁×#499	2
C₃₁×#482	2
C₃₁×#481	2
C₃₁×#478	2
C₃₁×#466	2
C₄×D₈×C₃₁	2
C₈×D₄×C₃₁	2
C₃₁×#427	2
C₂×C₃₁×#107	2
#3339	31

Subgroups — C31×#3339 #16843272

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3339 #16843272